Synchronizing Strongly Connected Partial DFAs

Authors Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, Marek Szykuła

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Mikhail V. Berlinkov
  • Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg, Russia
Robert Ferens
  • Institute of Computer Science, University of Wrocław, Poland
Andrew Ryzhikov
  • Université Gustave Eiffel, LIGM, Marne-la-Vallée, France
Marek Szykuła
  • Institute of Computer Science, University of Wrocław, Poland


We are grateful to the anonymous reviewers for useful comments and recent literature references.

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Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, and Marek Szykuła. Synchronizing Strongly Connected Partial DFAs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a reset word) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs with the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Mathematics of computing → Combinatorics
  • Černý conjecture
  • literal automaton
  • partial automaton
  • prefix code
  • rank conjecture
  • reset threshold
  • reset word
  • synchronizing automaton
  • synchronizing word


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